REVISTA MATEMATICA de la Universidad Complutense de Madrid, Vol 4, nú 2 y 3, 1991.
publications
(1) The Lie algebroid of a transversally complete foliation was constructed by P.Molino in 1977. Since each foliation of left cosets of G by a nonclosed Lie subgroup H is transversally complete, we obtain in this way a transitive Lie algebroid A(G,H) of this foliation. The definition of A(G, H) independent of the theory of P.Molino is given in this work. The following structures theorems concerning A(G, H) are obtained:
- The adjoint Lie algebra bundle of A(G, H) is a trivial bundle of abelian Lie algebras,
- If the Lie algebroid A(G, H) admits a flat connection, then it is trivial.
(2) A Lie subalgebra h of a Lie algebra g is said to be minimally closed (after A.Malcev) if the corresponding connected Lie subgroup is closed in the simply connected Lie group determined by g. The aim of this paper is to prove the following theorem:
- Let HÌG be any connected (not necessarily closed) Lie subgroup of a Lie group G. Denote by h, h¯ and g the Lie algebras of H, of its closure H¯ and of G, respectively. If there exists a Lie subalgebra cÌg such that c+h¯=g, cÇh¯=h, then A(G, H) admits a flat connection and h is minimally closed.
As a corollary we obtain that if p1(G) is finite, then no such a Lie subalgebra c exists provided that H is nonclosed.